• Tidak ada hasil yang ditemukan

Paper19-Acta26-2011. 122KB Jun 04 2011 12:03:16 AM

N/A
N/A
Protected

Academic year: 2017

Membagikan "Paper19-Acta26-2011. 122KB Jun 04 2011 12:03:16 AM"

Copied!
10
0
0

Teks penuh

(1)

ON P-VALENT STRONGLY STARLIKE AND STRONGLY CONVEX FUNCTIONS

Ali muhammad

Abstract. In this paper, we define some new classes S(a, c, λ, p, α, β) and

C(a, c, λ, p, α, β) of strongly starlike and strongly convex functions or order αand type β by using Cho- Kown -Srivastava integral operators. We also derive some interesting properties, such as inclusion relationships of these classes.

2000Mathematics Subject Classification: 30C45, 30C50.

Keywords: Analytic functions; Strongly srarlike functions; Strongly convex func-tions; Cho Kown Srivastava integral operators.

1. Introduction

LetA(p) denote the class of functionsf(z) normalized by

zp+ ∞

X

k=1

ap+kzp+k, p∈N={1,2,3, ...}. (1.1)

which are analytic and p-valent in the unit diskE ={z:z∈Cand |z|<1}.

A function f(z) ∈A(p) is said to be in the class S∗(p, β) of p-valently starlike function of order β inE if it satisfies the following inequality:

Re

zf′(z)

f(z)

> β, z∈E, 0≤β < p. (1.2) The class S∗(p, β) was introduced and studied by Goodman [4], see also [11]. For recent investigation and more detail the interested reader are refers to the work by [12, 13]. Also, we note thatS∗(p,0) =S

p,whereSp∗is the class of p-valently starlike

(2)

A function f(z) ∈ A(p) is said to be in the class C(p, β) of p-valently convex function of order β inE if it satisfies the following inequality:

Re

1 +zf ′′(z)

f′(z)

> β, z∈E, 0≤β < p. (1.3) The class C(p, β) was introduced and studied by Goodman [4], see also [10]. Also we note that C(p,0) = Cp, where Cp is the class of p-valently convex functions in

E.

A function f(z) ∈A(p) is said to be strongly starlike of order α and type β in

E if it satisfies the following inequality:

arg

zf′(z)

f(z) −β

< απ

2 , z ∈E, 0≤α < pand 0≤β < p. (1.4) We denote this class by S∗(p, α, β).A function f(z) A(p) is said to be strongly convex of order α and type β inE if it satisfies the following inequality:

arg

1 +zf ′′(z)

f′(z) −β

< απ

2 , z ∈E, 0≤α < pand 0≤β < p. (1.5) We denote this class by C(p, α, β).

It is obvious that f(z) ∈ A(p) belongs to C(p, α, β) if and only if zff(z(z)) ∈

S∗(p, α, β).Also, we note thatS(p, α, β) =S(p, β) and C(p, α, β) =C(p, β). In our present investigation we shall make use of the Gauss hypergeometric functions defined by

2F1(a, b;c;z) = ∞

X

k=0

(a)k(b)kzk

(c)k (1)k

, (1.6)

wherea, b, c∈C,c /Z−

0 ={0,−1,−2, ...}and (k)ndenote the Pochhammer symbol

(or the shifted factorial) given, in terms of the Gamma function Γ, by (k)n=

Γ(k+n) Γ (k) =

k(k+ 1)...(k+n−1) n∈N,

1 n= 0.

We note that the series defined by (1.6) converges absolutely for z ∈E and hence represents an analytic function in the open unit disk E, see [14].

We define a functionφp(a, c;z) by

φp(a, c;z) =zp+ ∞

X

k=1

(a)k zp+k

(c)k

, a∈R; cR\Z−

(3)

Using the function φp(a, c;z),we consider a function φ†p(a, c;z) defined by

φp(a, c;z)∗φ†p(a, c;z) = z

p

(1−z)λ+p, z ∈E,

where λ >−p.This function yields the following family of linear operators

Ipλ(a, c)f(z) =φ†p(a, c;z)∗f(z), z∈E, (1.7) wherea, c∈R\Z−

0. For a functionf(z)∈A(p), given by (1.1), it follows from (1.7) that for λ >−p and a, c∈R\Z−

0

Ipλ(a, c)f(z) = zp+ ∞

X

k=1

(a)k (λ+p)

(c)k (1)k

ap+kzp+k (1.8)

= zP2F1(c, λ+p;a;z)∗f(z), z∈E. From equation (1.8) we deduce that

z(Ipλ(a, c)f(z))´= (λ+p)Ipλ+1(a, c)f(z)−λIpλ(a, c)f(z), (1.9)

z(Ipλ(a+ 1, c)f(z))´=aIpλ(a, c)f(z)−(a−p)Ipλ(a+ 1, c)f(z). (1.10) We also note that

I0

p(p+ 1,1)f(z) =p z R

0

f(t)

t dt, Ip0(p,1)f(z) =Ip1(p+ 1,1)f(z) =f(z), Ip1(p,1)f(z) = zfp′(z), Ip2(p,1)f(z) = 2zf′(pz()+p+1)z2f′′(z),

Ip2(p+ 1,1)f(z) =

f(z)+zf′(z)

p+1 , I

n

p(a, a)f(z) =Dn+p−1f(z), n∈N, n >−p,

where Dn+p−1is the Ruscheweyh derivative of (n+p1)thorder, see [5]. The operator Iλ

p(a, c) (λ > −p, a, c ∈ R\Z−0) was recently introduced by Cho et.al [1], who investigated (among other things) some inclusion relationships and argument properties of various subclasses of multivalent functions in A(p), which were defined by means of the operator Iλ

p(a, c).

Forλ=c= 1 and a=n+p,Cho-kown-Srivastava operator Iλ

p(a, c) yields Ip1(n+p,1) =In,p(n >−p),

(4)

Using the operatorIλ

p(a, c) we now define new subclasses of A(p) as follows:

Definition 1.1.

S∗(a, c, λ, p, α, β) =

(

f(z)∈A(p) :Ipλ(a, c)f(z)∈S∗(p, α, β), z(Iλ

p(a, c)f(z))′ Iλ

p(a, c)f(z)

6=β )

.

where z∈E.

It is easy to see thatS∗(p,1,0, p, α, β) =S(p, α, β) and S(p+ 1,1,1, p, α, β) =

S∗(p, α, β),is the class of strongly starlike functions of orderα and type β as given by (1.4).

Definition 1.2.

C(a, c, λ, p, α, β) =

(

f(z)∈A(p) :Ipλ(a, c)f(z)∈C(p, α, β),1 +z(I

λ

p(a, c)f(z))′′

(Iλ

p(a, c)f(z))′

6=β )

.

where z∈E.

Obviously, f(z) ∈ C(a, c, λ, p, α, β) if and only if zfp′(z) ∈ S∗(a, c, λ, p, α, β),

C(p,1,0, p, α, β) = C(p, α, β) and C(p + 1,1,1, p, α, β) = C(p, α, β) is the class of strongly convex functions of order α and type β as given by (1.5).

2. Preliminaries and Main results

In order to prove our main results we shall need the following result.

Lemma 2.1.[9] Let a function h(z) = 1 +c1z+c2z2+...be analytic in E and

h(z)6= 0, z∈E.If there exists a point z0 ∈E such that |argh(z)|< π

2α (|z|<|z0|) and |argh(z0)|=

π

2α (0≤α <1), (2.1) then we have,

z0h′(z0)

h(z0)

=ikα, (2.2)

where

k≥ 1 2(a+

1

a) when argh(z0) = π

2α, (2.3)

k≤ −1 2((a+

1

a) when argh(z0) = π

2α, (2.4)

and

(h(z0))

1

(5)

Theorem 2.2. S∗(a, c, λ+1, p, α, β)S(a, c, λ, p, α, β)S(a+1, c, λ, p, α, β). Proof. Set

z(Iλ

p(a, c)f(z))′ Iλ

p(a, c)f(z)

=β+ (p−β)h(z). (2.5) Thenh(z) is analytic inE withh(0) = 1 andh(z)6= 0, z ∈E.Applying the identity (1.9) in (2.1) and differentiating the resulting equation with respect to z we have

z(Iλ+1(a, c)f(z))

Iλ+1(a, c)f(z) −β = (p−β)h(z) +

(p−β)zh′(z) (λ+β) + (p−β)h(z).

Suppose there exists a pointz0∈Esuch that the conditions (2.1) to (2.4) of Lemma 2.1 are satisfied. Thus, if

argh(z0) = −π

2 α, z0 ∈E, then

z(Iλ+1(a, c)f(z))

Iλ+1(a, c)f(z) −β = (p−β)h(z0)

1 +

z0h′(z0)

h(z0)

(λ+β) + (p−β)h(z0)

= (p−β)aαe−πα2

1 + ikα

(λ+β) + (p−β)aαe−πα 2

.

This implies that arg

z(Iλ+1(a, c)f(z))

Iλ+1(a, c)f(z) −β

= −πα 2 + arg

1 + ikα

(λ+β) + (p−β)aαe−πα2

= −πα 2

+ tan−1

"

kα((λ+β) + (p−β)aαcos πα

2

(λ+β)2+ 2(λ+β)(pβ)aαcos πα

2

+ (p−β)2a(pβ)aαsin πα

2

#

.

This gives that

arg

z(Iλ+1(a, c)f(z))′

Iλ+1(a, c)f(z) −β

≤ −πα 2 , since

k≤ −1 2 (a+

1

a)≤ −1 andz0 ∈E,

this contradicts the condition f(z)∈S∗(a, c, λ, p, α, β). On the other hand if we set

argh(z0) =

πα

(6)

then it can similarly be shown that arg

z(Iλ+1(a, c)f(z))

Iλ+1(a, c)f(z) −β

≥ πα 2 , since

k≥ 1 2(a+

1

a) and z0 ∈E,

which again contradicts the hypothesis thatf(z)∈S∗(a, c, λ, p, α, β).

Thus the function defined by (2.5) has to satisfy the following inequality: argh(z)≤ πα

2 , z∈E. which implies that

arg

z(Iλ+1(a, c)f(z))

Iλ+1(a, c)f(z) −β

< πα

2 , z ∈E.

The proof of part (ii) lies on the similar lines. This completes the proof of Theorem 2.2.

Theorem 2.3.C(a, c, λ+ 1, p, α, β)⊂C(a, c, λ, p, α, β)⊂C(a+ 1, c, λ, p, α, β) Proof. To prove this inclusion relationship, we observe from Theorem 2.2 that

f(z) ∈ C(a, c, λ+ 1, p, α, β)⇔ zf ′(z)

p ∈S

(a, c, λ+ 1, p, α, β) ⇒ zf

(z)

p ∈S

(a, c, λ, p, α, β)f(z)C(a, c, p, α, β).

The proof of second part lies on similar lines. This completes the proof of Theorem 2.3.

For a functionf(z)∈A(p) the integral operator,Fδ,p:A(p)−→A(p) is defined

by

Fδ,p(f)(z) =

δ+p zp

z Z

0

tδ−1f(t)dt= zp+ ∞

X

k=1

δ+p δ+p+kz

p+k !

∗f(z) (2.6) = zp 2F1(1, δ+p, δ+p+ 1;z)∗f(z), z∈E, see [2].

It follows from (2.6) that

z(Ipλ(a, c)Fδ,p(f)(z) ′

(7)

We have the following result.

Theorem 2.4. Let δ >−β and 0≤β < p.If f(z)∈ S∗(a, c, λ, p, α, β) with

z(Iλ

p(a, c)Fδ,p(f)(z))′ Iλ

p(a, c)Fδ,p(f)(z)

6=β, for allz∈E,

then we have

Fδ,p(f)(z)∈S∗(a, c, λ, p, α, β).

Proof. We begin by setting

z( Iλ

p(a, c)Fδ,p(f)(z))

p(a, c)Fδ,p(f)(z)

=β+ (p−β)h(z), z∈E. (2.8) Then h(z) is analytic in E with h(0) = 1.Using (2.7) and (2.8), we find that

(δ+p) (I

λ

p(a, c)f(z) Iλ

p(a, c)Fδ,p(f)(z)

= (δ+p) + (p−β)h(z). (2.9) Differentiation both sides of (2.9) logarithmically, we obtain

z( Iλ

p(a, c)f(z))′ Iλ

p(a, c)f(z)

−β = (p−β)h(z) + (p−β)zh ′(z) (δ+β) + (p−β)h(z). Suppose now there exists a point z0∈E such that

|argh(z)|< π

2α (|z|<|z0|) and |argh(z)|=

π

2α. Then by Lemma 2.1, we can write that

z0h′(z0)

h(z0)

=ikα and (h(z0))α1 =±iα (α >0).

If

argh(z0) =

π

2α, z0 ∈E, then

z(Iλ

p(a, c)f(z0))′

p(a, c)f(z0)

−β = (p−β)h(z0)

1 +

zh′(z0)

h(z0)

(δ+β) + (p−β)h(z0)

= (p−β)aαeiπα2

1 + ikα

(δ+β) + (p−β)aαeiπα2

(8)

This shows that arg z(I

λ

p(a, c)f(z0))′

p(a, c)f(z0) −β

!

= πα 2 + arg

1 + ikα

(δ+β) + (p−β)aαeiπα 2

= πα 2

+ tan−1

(

kα(δ+β+ (p−β))aαcos πα

2

(δ+β)2+ 2(δ+β)(pβ)aαcos πα

2

+ (p−β)2a+(pβ)aαsin πα

2 ) ≥ πα 2 , since

k≥ 1 2(a+

1

a)≥1 and z0 ∈E,

which contradicts the assumption that f(z)∈S∗(a, c, λ, p, α, β).

Similarly in the case when

argh(z0) =−

π

2α, z0 ∈E, we can prove that

arg z(I

λ

p(a, c)f(z0)) ′

p(a, c)f(z0) −β

!

≤ −πα 2 , since

k≤ −1 2(a+

1

a)≤ −1 andz0 ∈E,

contradicting once again the condition thatf(z)∈S∗(a, c, λ, p, α, β). We thus conclude that h(z) must satisfy the following inequality

|argh(z)|< π

2α, z∈E. This shows that

argz(I

λ

p(a, c)Fδ,p(f)(z))′ Iλ

p(a, c)Fδ,p(f)(z)

−β < πα

2 , z∈E, and hence the proof is complete.

Theorem 2.5. Let δ >−β and 0≤β < p.If f(z)∈C(a, c, λ, p, α, β) and

1 +z(I

λ

p(a, c)Fδ,p(f)(z))′′

(Iλ

p(a, c)Fδ,p(f)(z))′

(9)

then we have

Fδ,p(f)(z)∈C(a, c, λ, p, α, β).

Proof. Since

f(z) ∈ C(a, c, λ, p, α, β)⇔ zf ′(z)

p ∈S

(a, c, λ, p, α, β) ⇒ Fδ,p(f)(z)

p ∈S

(a, c, λ, p, α, β) z(Fδ,p(f)(z))′

p ∈S

(a, c, λ, p, α, β) ⇔ Fδ,p(f)(z)∈C(a, c, λ, p, α, β).

Acknowledgements. We would like to thank Honourable Director Finance Sarwar Khan for providing us excellent research facilities.

References

[1] N. E. Cho, O. S. Kown, and H. M. Srivastava, Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators, J. Math. Anal. App. 292 (2004), 470 -483.

[2] J. H. Choi, M. Siago, and H .M. Srivastava, Some inclusion properties of certain family of integral operators, J. Math. Anal. Appl. 276 (2002), 432 - 445.

[3] N. E. Cho, O. H. Kown, and H. M. Srivastava,Inclusion and argument prop-erties for certain subclasses of meromorphic functions associated with a family of multiplier transformations, J. Math. Anal. Appl. 300 (2004), 505- 520.

[4] A. W. Goodman, On the Schwarz Christoffel transformation and P-valent functions, Trans. Amer. Math. Soc. 68 (1950), 204 - 233.

[5] R. M. Goel and N. S. Sohi, A new criterion for p-valent functions, Proc. Amer. Math. Soc. 78 (1980), 353 - 357.

[6] J. L. Liu and K. I. Noor,Some properties of Noor integral operator, J. Natur. Geom. 21 (2002),81-90.

[7] K. I. Noor, Some classes of p-valent analytic functions defined by certain integral operator, Appl. Math. Compute. 157 (2004), 835-840.

[8] K. I. Noor and M. A. Noor, On certain classes of analytic functions defined by Noor integral operator, J. Math. Appl. 281 (2003), 244-252.

[9] M. Nunokawa, On the order of strongly starlikness of strongly convex func-tions, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 234-237.

(10)

[11] D. A. Patel and N. K. Thakare, On convex hulls and extreme points of P-valent starlike and convex classes with applications, Bull. Math. Soc. Sci. Math. Roumanie (N.S) 27 (1983), 145 -160.

[12] J. Patel, N. E. Cho and H. M. Srivastava,Certain subclasses of multivalent functions associated with a family of linear operators, Math. Comput. Modelling 43 (2006), 320-338.

[13] H. M. Srivastava, J. Patel, and G. P. Mohapatra,A certain class of p-valently analytic functions, Math. Comput. Modelling 41 (2005), 321-334.

[14] E. T. Whittaker and G. N. Watson,A course on Modern Analysis: An intro-duction to the general Theory of Infinite processes and of Analytic Functions; With an account of the principle of Transcendental Functions, 4th Edition, Cambridge University Press, Cambridge, 1927.

Ali Muhammad

Depatment of Basic Sciences

University of Engineering and Technology Technology Peshawar Pakistan

Referensi

Dokumen terkait

llal-lrel yang telum iehs dapat ditanyakan pada saat pendaftaran. Ilemikhn Filgumunan ini, atas pertratiannya kami sampa*an

Berdasarkan Surat Penetapan Pemenang Pelelangan Pelaksana Pekerjaan Pengadaan Alat Kesehatan Puskesmas, Kegiatan Pengadaan Sarana Kesehatan (DPPID) Dinas

Dari hasil penelitian dapat diketahui bahwa keselamatan kerja secara parsial tidak berpengaruh secara signifikan terhadap komitmen organisasional dengan indikator

Unit Layanan Pengadaan Kota Banjarbaru mengundang penyedia Pengadaan Barang/Jasa untuk mengikuti Pelelangan Umum pada Pemerintah Kota Banjarbaru yang dibiayai dengan Dana APBD

Budaya organisasional perusahaan dapat dilihat dari pengelolaan manajemen perusahaan yang melibatkan langsung pemilik perusahaan pada karyawan untuk memberikan penanganan

Selain itu, pemimpin juga memiliki beberapa karakter pemimpin Kristen dalam hubungan dengan sesama yaitu, ingin melayani orang lain, mau berbagi, mengampuni

Sedangkan menurut Daft (2011, p.153) personal moral development seseorang terdiri dari 3 level yaitu preconventional level, conventional level, dan postconventional

Pengalaman kerja bersifat sebagai full mediation variable (variabel mediasi penuh) dikarenakan pengaruh langsung praktek sumber daya manusia terhadap sikap perilaku